Theorem fssxp 3643

Description: A mapping is a class of ordered pairs.

Assertion

Ref	Expression
fssxp	|- (F:A-->B -> F (_ (A X. B))

Proof of Theorem fssxp

Step	Hyp	Ref	Expression
1		frn 3639	. . . 4 |- (F:A-->B -> ran F (_ B)
2		ssid 2083	. . . . 5 |- A (_ A
3		ssxp 3262	. . . . 5 |- ((A (_ A /\ ran F (_ B) -> (A X. ran F) (_ (A X. B))
4	2, 3	mpan 697	. . . 4 |- (ran F (_ B -> (A X. ran F) (_ (A X. B))
5	1, 4	syl 10	. . 3 |- (F:A-->B -> (A X. ran F) (_ (A X. B))
6		fdm 3637	. . . 4 |- (F:A-->B -> dom F = A)
7		xpeq1 3206	. . . 4 |- (dom F = A -> (dom F X. ran F) = (A X. ran F))
8		sseq1 2085	. . . 4 |- ((dom F X. ran F) = (A X. ran F) -> ((dom F X. ran F) (_ (A X. B) <-> (A X. ran F) (_ (A X. B)))
9	6, 7, 8	3syl 20	. . 3 |- (F:A-->B -> ((dom F X. ran F) (_ (A X. B) <-> (A X. ran F) (_ (A X. B)))
10	5, 9	mpbird 196	. 2 |- (F:A-->B -> (dom F X. ran F) (_ (A X. B))
11		frel 3636	. . 3 |- (F:A-->B -> Rel F)
12		relssdr 3519	. . 3 |- (Rel F -> F (_ (dom F X. ran F))
13		sstr2 2074	. . 3 |- (F (_ (dom F X. ran F) -> ((dom F X. ran F) (_ (A X. B) -> F (_ (A X. B)))
14	11, 12, 13	3syl 20	. 2 |- (F:A-->B -> ((dom F X. ran F) (_ (A X. B) -> F (_ (A X. B)))
15	10, 14	mpd 26	1 |- (F:A-->B -> F (_ (A X. B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   (_ wss 2050   X. cxp 3174  dom cdm 3176  ran crn 3177  Rel wrel 3181  -->wf 3184

This theorem is referenced by:  funssxp 3644  opelf 3646  fabexg 3659  dff4 3822  dff2 3823  fopabssxp 3830  mapex 4334  mapval2 4341  mapsspw 4347  uniixp 4363  infmap2 7583  lmbrf 7927  iscauf 7936  iscau5 7938  iscaunns 7941  lmclimnn 7961  h2hcau 8844  h2hlm 8845  1alg 10625

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200

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